The Ehrenfeucht-Fraïssé-game of length ₁

Let A and B be two first order structures of the same vocabulary. We shall consider the Ehrenfeucht-Fraisse-game of length ω 1 of A and B which we denote by G ω1 (A, B). This game is like the ordinary Ehrenfeucht-Fraisse-game of L ωω except that there are ω 1 moves. It is clear that G ω1 (A, B) is determined if A and B are of cardinality ≤ N 1 . We prove the following results: Theorem 1. If V = L, then there are models A and B of cardinality N 2 such that the game G ω1 (A, B) is nondetermined. Theorem 2. If it is consistent that there is a measurable cardinal, then it is consistent that G ω1 (A, B) is determined for all A and B of cardinality ≤ N 2 . Theorem 3. For any k ≥ N 3 there are A and B of cardinality k such that the game G ω1 (A, B) is nondetermined